47 problems

Let $\Theta^\infty=\Theta^{L(\Gamma^\infty,\mathbb{R})}$ and let $L(\mathcal{A}^\infty)$ be the model generated by $\mathcal{A}^\infty$. Square-sequence conjecture. … The claim is…

Can there exist a model $V$ of $\mathrm{ZFC}$ in which $\aleph_\omega$ is the least cardinal $\kappa$ such that $\mathcal{P}^V(\kappa)\neq\mathcal{P}^{\mathrm{HOD}^V}(\kappa)$; tha…

Let $\mathrm{EEA}$ denote the Extender Embedding Axiom, and let $V=\mathrm{Ultimate-}L$ denote the corresponding Ultimate-$L$ axiom. Woodin's Ultimate-$L$ conjecture. The theory ……

Let $\gamma$ be a limit ordinal, and let $(\mathrm{M}^{\alpha})_{\alpha\in\mathrm{Ord}}$ denote the sequence of inner mantles, with $\mathrm{M}^{\alpha}$ its stage at index…

HOD conjecture. There is a proper class of regular uncountable cardinals $\kappa$ which are not $\omega$-strongly measurable in $\operatorname{HOD}$.

Assume ${\sf{AD}}^+$. Let $\kappa$ be a regular Suslin cardinal, and let ${\mathbf{\Gamma}}$ be the pointclass of $\kappa$-Suslin sets. Suslin-cardinal unreachable successor conjec…

Let $A_{(\alpha,n)}$ be the sets defined from $J_{\alpha}(\mathbb{R})$, and let a mouse set mean a set $A\subseteq\mathbb{R}$ of the form $A=\mathbb{R}\cap\mathcal{M}$ for some cou…

Let $({\mathcal{P}},\Psi)$ be a hod pair such that … Let $\kappa$ be the least strong cardinal which reflects the class of strong cardinals, and let…

Let $C(\mathop{\text{aa}})$ and $C^2(\omega)$ denote the inner models defined in the paper, with $C(\mathop{\text{aa}})$ the model generated using the relevant class of structures…

Let $C^2(\alpha9\omega)$ denote the inner model obtained from second-order definability over $\alpha9\omega$, let $\text{CH}$ denote the continuum hypothesis, and let $M_n$ denote th…

Solovay-sequence conjecture. Then $N\vDash\mathsf{ZF}$, $\Theta^N=\theta_\alpha$, $N$ satisfies that every cardinal $\eta\geq\Theta$ has regular successor $\eta^+$, and, for every…

Cardinal-regularity conjecture. Then

Solovay-sequence conjecture. Under these assumptions, $V_\zeta^{{\sf{HOD}}_X}$ is a model of

Least-cardinal structural reflection conjecture. A cardinal $\kappa$ is the least cardinal satisfying some large-cardinal notion if and only if $\kappa$ is the least cardinal satisfy…

Assume $\mathrm{ZF}+\mathsf{AD}^{L(\mathbb R)}$. Let $M$ be a $(0,\omega_1)$-iterable countable $\omega$-small premouse satisfying “$\omega_1$ exists”. Then there are $\gamma$,…

Let $(\mathcal{V},\Omega)$ be an excellent least branch hod pair with $\mathcal{V}\models\mathsf{ZFC}$, let $\delta$ be a Woodin limit of Woodin cardinals in $\mathcal{V}$, and let…

Let $V$ satisfy $obreak\textsf{AD}_{\mathbb R}+ obreak\textsf{NLE}+V=L({\wp}(\mathbb R))$. Let $obreak\Delta\subseteq{\wp}(\mathbb R)$ and let…

Let $obreak\textsf{AD}_{\mathbb R}$ denote determinacy for real games. Suppose that for every set $B\subseteq\mathbb R$ there is a function $F$ as in Theorem nx such that $F$ is n…

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy. For a function $F$ as in the paper's Theorem nx, call $F$ pe-bad when, for a cone of reals $x$, the supremu…

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreak\textsf{NLE}$ denote the no-long-extender assertion. Let $obreak\textsf{HPC}$ denote hod-p…

Let $obreak\textsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreak\textsf{NLE}$ be the no-long-extender assertion. Let $obreak\textsf{LEC}$ denote…

Covering with Chang Models conjecture. The asserted transitive model $M$ exists with all the listed properties. This conjecture is intended to overcome limitations of the core mode…

Assume $ZF + AD + DC + V=L(\mathbb{R})$. Let $\alpha\in ON$ and $n\in\omega$. For each $x\in\mathbb{R}$, define $\Gamma_x$ to be the collection of all pointsets $A$ such that for s…

Assume $ZF + AD + DC + V=L(\mathbb{R})$. A mouse set is a set $X\subseteq\mathbb{R}$ for which there is an $\omega_1+1$-iterable premouse $M$ with $X=M\cap\mathbb{R}$. Let $\Gamma$…

Let $LSA$ abbreviate the assertion that $AD^+$ holds and there is a largest Suslin cardinal belonging to the Solovay sequence. Let $\vec{E}$ be an extender sequence and write…